Proving Irrationality: Is sqrt2 + sqrt5 + sqrt7 Rational or Irrational?

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In summary, irrationality refers to numbers that cannot be expressed as a simple fraction or ratio of two integers. Rational numbers can be expressed as a fraction, while irrational numbers cannot. The sum of square roots, sqrt2 + sqrt5 + sqrt7, is an irrational number and can be proven by using algebraic numbers and properties of rational and irrational numbers. This proof is important for understanding the nature of numbers and has practical applications in various fields.
  • #1
hew
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Homework Statement



decide if sqrt2 + sqrt5 +sqrt7 is rational or irrational, with proof

Homework Equations





The Attempt at a Solution


i assumed that it was rational, the equation =r
sqrt2 + sqrt 5=r- sqrt7
7 + 2sqrt10=r^2 + 7 -2r*sqrt7
2sqrt10=r^2 -2r*sqrt7
could i have a hint on how to rearrange this, i have already proved in an earlier question that sqrt2, sqrt5, sqrt7 are irrational

thankyou
 
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  • #2
You're almost there. square both sides again, and you can write sqrt(7) as a linear combination of rational numbers. i.e. you've shown sqrt(7) is rational.
 
  • #3


To prove whether sqrt2 + sqrt5 + sqrt7 is rational or irrational, we can use the proof by contradiction method.

Assuming that sqrt2 + sqrt5 + sqrt7 is rational, we can represent it as a/b, where a and b are integers and b is not equal to 0.

Now, we can rearrange the given equation as sqrt2 + sqrt5 = (r - sqrt7), where r is a rational number.

Squaring both sides, we get 2 + 2sqrt10 + 5 = r^2 + 7 - 2rsqrt7

Rearranging, we get 2sqrt10 = r^2 - 2rsqrt7

Squaring both sides again, we get 40 = r^4 + 4r^2*7 - 4r^3sqrt7

Since r^4 and 4r^2*7 are rational numbers, in order for the equation to hold, 4r^3sqrt7 must also be rational.

However, we already know that sqrt7 is irrational from an earlier question. Therefore, our assumption that sqrt2 + sqrt5 + sqrt7 is rational must be false.

Hence, we can conclude that sqrt2 + sqrt5 + sqrt7 is irrational.
 

1. What is irrationality?

Irrationality refers to a number that cannot be expressed as a simple fraction or ratio of two integers. It is a decimal with infinite non-repeating digits. Examples of irrational numbers include pi (3.14159...) and the square root of 2 (1.414213...).

2. What is the difference between rational and irrational numbers?

Rational numbers can be expressed as a simple fraction or ratio of two integers, while irrational numbers cannot. Rational numbers have either a finite or repeating decimal representation, while irrational numbers have an infinite non-repeating decimal representation.

3. Is sqrt2 + sqrt5 + sqrt7 a rational or irrational number?

The sum of square roots, sqrt2 + sqrt5 + sqrt7, is an irrational number. This can be proven by assuming the opposite, that it is a rational number, and showing that it leads to a contradiction. Therefore, it must be irrational.

4. How can we prove the irrationality of sqrt2 + sqrt5 + sqrt7?

The proof of the irrationality of sqrt2 + sqrt5 + sqrt7 involves using the concept of algebraic numbers and the properties of rational and irrational numbers. In short, it involves showing that the sum of square roots results in a number that cannot be expressed as a ratio of two integers, making it an irrational number.

5. Why is proving the irrationality of sqrt2 + sqrt5 + sqrt7 important?

The proof of the irrationality of sqrt2 + sqrt5 + sqrt7 is important because it helps us better understand the nature of numbers and their properties. It also has applications in various fields such as mathematics, physics, and engineering. Additionally, this proof can serve as a basis for proving the irrationality of other numbers and further expanding our knowledge of irrational numbers.

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